Time allowed: 20 minutes.
1. (2 marks)
Suppose
P, Q, R
are invertible matrices of the same size.
(a) Simplify the following expression, giving brief reasons for each step :
P
3
(
RP
2
)
−
1
(
Q
3
R
−
1
)
−
1
Q
2
−
Q
−
1
P
(b) If det(
R
) =
1
2
,
det(
P
) = 3 and det(
P
2
Q
2
R
−
1
) = 2
,
then find det(
Q
).
2. (2 marks)
Three companies
X, Y
and
Z
reinsure with one another to share their risk:
•
For each $1 that
X
pays out in claims,
X
receives $0.30 from
Y
and $0.10 from
Z
.
•
For each $1 that
Y
pays out in claims,
Y
receives $0.20 from
X
and $0.10 from
Z
.
•
For each $1 that
Z
pays out in claims,
Z
receives $0.20 from
X
and $0.10 from
Y
.
In a particular year, the total cost to
X
(claims paid and amounts paid to
Y
and
Z
, less amounts received
from
Y
and
Z
) was $26 million.
Similarly, the total cost to
Y
was $22 million and the total cost to
Z
was $20 million.
Determine the total claims paid out by companies
X
,
Y
and
Z
.
3. (2 marks)
Suppose that

˜
a

=

˜
b

. Show that
˜
a
+
˜
b
and
˜
a
−
˜
b
are orthogonal.
4. (4 marks)
Consider the matrix
P
=
1
−
p
q
p
1
−
q
where 0
< p <
1 and 0
< q <
1. Define the matrix
S
by
S
=
P
−
I
, where
I
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 Three '11
 Invertible matrix, $0.20, rn S

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